Astronomy & Celestial Mechanics
the sky, computed
Concept

Choreography — where it appears

A periodic solution of the many-body problem in which every body travels the same closed curve, each a fixed fraction of a period behind the next. The figure-eight of three equal masses is the best known and is proved to exist.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Three bodies, integrated. Three equal masses integrated forward under mutual gravity: the figure-eight choreography — all three bodies on one closed curve. Every point is a step of the equations of motion, and the total energy is conserved to 2.5e-11 across the run.

Three bodies, and what "no solution" actually means

The three-body problem is routinely called unsolvable. Trajectories are computed for it every day, exact periodic solutions are known, and both statements are true — the word is doing more work than it looks.

gravitation · The three-body problem The figure-eight with equal masses, and with one mass changed. The paths of three bodies in the figure-eight choreography over 100 periods, with equal masses (left) and with the third body's mass changed by 0.0001 (right). With equal masses the three paths lie on one another: each body runs round the same figure-of-eight, a third of a period behind the next. With the mass changed the three bodies stay bound but leave the single curve, filling a broad band round where it was. The choreography is an exact solution only for exactly equal masses, and the island of stable orbits near it is narrow.

A periodic orbit that the sky cannot keep

Three equal masses can chase one another for ever round a single figure of eight. The orbit exists, it is stable, and it will almost certainly never be seen — because its island of stability is a hundredth of a per cent wide in the masses, and nothing in nature makes three stars equal to that precision.

gravitation · The three-body problem

Named alongside it

The objects these essays reach for when they reach for this one.

Three-body problemAngular momentumEccentricityEffective potentialFigure-eight orbitIntegrabilityIsland of stabilityKAM theoryLagrange pointsLinear stabilityLyapunov timeMass ratio

All concepts